Exchange, Reciprocity and Stability
Revisiting mathematical criteria for stable coexistence
I’d been puzzled for a while about microbial interactions.
Ecologists often think about interactions in terms of direct effect of one species population on another—the Lotka-Volterra equations for competition are a classic example . If you and I compete for a common resource, these equations tell us that as my population size gets larger, your growth rate will get smaller. And vice versa. The picture can easily be extended to model mutualistic interactions by flipping a few parameters from negative to positive, and in recent years many research labs have modeled microbial interactions in this framework (e.g. [2,3]).
That process of actually fitting parameter values for pairwise interactions is challenging, whether it's in microbial systems or elsewhere. On the other hand, by assuming that interaction strengths are randomly drawn from a distribution of possible values, and leveraging the mathematics of random matrices, ecologists have gained very general insights about equilibria and stability for pairwise interactions . Morally speaking, these results have told us that a complex system with too many, or too strong interactions will tend not to have a stable equilibrium for species abundances—any small perturbation would send species abundances away from this equilibrium, and possibly towards collapse. Bad news for coexistence, and particularly bad for communities with a lot of mutualistic interactions, which (it turns out) are even less likely to have stable equilibria .
So this is where the puzzle started. I wondered how mutualistic interactions could be widespread in microbial communities , if at the same time this kind of interaction would push the system towards instability . To get a handle on this, I partnered with lab member Stacey Butler, a graduate student in the mathematics department at Illinois. Stacey and I had already been collaborating on various other projects, and this one nicely matched our shared interests in the mathematics of complex systems. So we started to think about a different framework for modeling interactions, where we explicitly considered the dynamics of the resources that microbial taxa consume and exchange. Consumer-resource frameworks also go back a very long way in ecology , but we couldn’t find the results we wanted to when exchange was considered alongside consumption.
I’ll highlight one of the new results that Stacey and I derived , that was a bit of a surprise: community stability is guaranteed in the case that all pairs of species exchange resources reciprocally, no matter how strong those mutualistic interactions are. It’s quite a different criterion than the classical results for community stability. We don’t know yet how this will extend to more general models of consumption and exchange—for example, our resource exchange is modeled on recycling of materials, and depends on current cell density, rather than on current population growth rate. We also don’t know how these results will interface with the evolution of resource exchange over time. And while understanding equilibria and stability is a starting point, we of course don’t know how close microbial communities even are to equilibria. So our paper settled some of our initial questions---but likely opens the door to a few more.
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